Quantum Field Theory and Gravity: Conceptual and Mathematical ...

348 D. Giulini
taken along different worldlines and intercompared by exchange of electromagnetic
signals. Suppose a field of light rays intersect the timelike worldlines
γ 1,2 of two clocks, the four-velocities of which are u 1,2 . Then the ratio of the
instantaneous frequencies measured at the intersection points of one integral
curve of k with γ 1 and γ 2 is
ν 2
= g(u 2,k)| γ2
. (2.5)
ν 1 g(u 1 ,k)| γ1
Note that this does not distinguish between gravitational and Doppler shifts,
which would be meaningless unless a local notion of “being at rest” were
introduced. The latter requires a distinguished timelike vector field, as e.g.
in stationary spacetimes with Killing field K. Then the purely gravitational
part of (2.5) is given in case both clocks are at rest, i.e. u 1,2 = K/‖K‖| γ1,2 ,
where γ 1,2 are now two different integral lines of K and ‖K‖ := √ g(K, K):
ν 2
:= g( k, K/‖K‖ ) √
| γ2
ν 1 g ( k, K/‖K‖ ) g(K, K)| γ1
=
. (2.6)
| γ1
g(K, K)| γ2
The last equality holds since g(k, K) isconstantalongtheintegralcurvesof
k, sothatg(k, K)| γ1 = g(k, K)| γ2 in (2.6), as they lie on the same integral
curve of k. Writingg(K, K) =:1+2U/c 2 and assuming U/c 2 ≪ 1, we get
Δν
ν := ν 2 − ν 1
= − U 2 − U 1
. (2.7)
ν 1 c 2
Possible deviations from this result are usually parametrised by multiplying
the right-hand side of (2.7) with (1 + α), where α = 0 in GR. In case of
violations of UCR/UGR, α may depend on the space-time point and/or on
the type of clock one is using. The lowest upper bound on α to date for comparing
(by electromagnetic signal exchange) clocks on different worldlines
derives from an experiment made in 1976 (so-called “Gravity Probe A”) by
comparing a hydrogen-maser clock in a rocket, that during a total experimental
time of 1 hour and 55 minutes was boosted to an altitude of about
10 000 km, to a similar clock on the ground. It led to [29]
α RS < 7 × 10 −5 . (2.8)
The best relative test, comparing different clocks (a 199 Hg-based optical clock
and one based on the standard hyperfine splitting of 133 Cs) along the (almost)
same worldline for six years gives [8]
α CR < 5.8 × 10 −6 . (2.9)
Here and above “RS” and “CR” refer to “redshift” and “clock rates”, respectively,
a distinction that we prefer to keep from now on in this paper, although
it is not usually made. To say it once more: α RS parametrises possible violations
of UGR by comparing identically constructed clocks moving along
different worldlines, whereas α CR parametrises possible violations of UCR by
comparing clocks of different construction and/or composition moving more
or less on the same worldline. An improvement in putting upper bounds on